A283526 - OEIS

%I #32 Nov 21 2024 21:26:32

%S 1,2,3,4,5,16,17,256,257,65536,65537,4294967296,4294967297,

%T 18446744073709551616,18446744073709551617,

%U 340282366920938463463374607431768211456,340282366920938463463374607431768211457

%N Pierce expansion of the number Sum_{k >= 1} 1/(2^(2^k - 1)).

%C This sequence is the Pierce expansion of the number 2*s(2) - 1 = 0.632843018043786287416159475061... where s(u) = Sum_{k>=0} 1/u^(2^k) that has been considered by J. Shallit in A007400. The continued fraction expansion of this number is essentially A006466.

%H Michael De Vlieger, <a href="/A283526/b283526.txt">Table of n, a(n) for n = 0..24</a>

%H Jeffrey Shallit, <a href="http://dx.doi.org/10.1016/0022-314X(79)90040-4">Simple continued fractions for some irrational numbers</a>. J. Number Theory 11 (1979), no. 2, 209-217.

%F a(0) = 1, a(2k+1) = 2^(2^k), a(2k+2) = 2^(2^k) + 1.

%e The Pierce expansion of 0.6328430180437862 starts as 1 - 1/2 + 1/(2*3) - 1/(2*3*4) + 1/(2*3*4*5) - 1/(2*3*4*5*16) + ...

%p L:=[1]: for k from 0 to 6 do: L:=[op(L),2^(2^k),2^(2^k)+1]: od: print(L);

%t {1}~Join~Map[{#, # + 1} &, 2^2^Range[0, 8]] // Flatten (* _Michael De Vlieger_, Mar 18 2017 *)

%Y Cf. A006466, A007400, A076214.

%K nonn

%O 0,2

%A _Kutlwano Loeto_, Mar 10 2017